Some branching formulas for Kac--Moody Lie algebras

In this paper we give some branching rules for the fundamental representations of Kac--Moody Lie algebras associated to $T$-shaped graphs. These formulas are useful to describe generators of the generic rings for free resolutions of length three described in [JWm16]. We also make some conjectures about the generic rings

Young tableaux, canonical bases and the Gindikin-Karpelevich formula

A combinatorial description of the crystal B(infinity) for finite-dimensional simple Lie algebras in terms of certain Young tableaux was developed by J. Hong and H. Lee. We establish an explicit bijection between these Young tableaux and canonical bases indexed by Lusztig's parametrization, and obtain a combinatorial rule for expressing the Gindikin-Karpelevich formula as a sum over the set of Young tableaux.Comment: 19 page

Vertex Operator Algebras Associated to Type G Affine Lie Algebras

In this paper, we study representations of the vertex operator algebra $L(k,0)$ at one-third admissible levels $k= -5/3, -4/3, -2/3$ for the affine algebra of type $G_2^{(1)}$. We first determine singular vectors and then obtain a description of the associative algebra $A(L(k,0))$ using the singular vectors. We then prove that there are only finitely many irreducible $A(L(k,0))$-modules from the category $\mathcal O$. Applying the $A(V)$-theory, we prove that there are only finitely many irreducible weak $L(k,0)$-modules from the category $\mathcal O$ and that such an $L(k,0)$-module is completely reducible. Our result supports the conjecture made by Adamovi{\'c} and Milas in \cite{AM}.Comment: 30 page